Background
In this example, we will show how to use the Emulator to prepare ordered ground states in Rydberg systems. The example is based on experimental works in 1D system, and 2D system.
Due to the strong Rydberg interactions, only one Rydberg excitation is allowed within the so-called blocakde radius Blockade. With positive detunings, more Rydberg excitations, however, are favored to lower the energies of the ground states. With the interplay of these two mechanisms, different ordered states are supported depending on the strength of blockade radius and the detuning. For example, in 1D there exists the $Z_N$ ordered states, and in 2D there are the checkboard phase, star phase, and even pure quantum phase (straited phase).
The adibatic process works as follows: One starts from all the atoms being in the ground state, which is also the ground states of many-body Hamiltonian with large negative detuning. Then the Rabi freuency $\Omega$ is turned on, and the detuning strength is ramped up from large negative value to postive values. If such process is slow enough, the quantum state the system stays close to the ground state of the instantaneous Hamiltonian. At the end of the process, one arrives a target Hamiltonian, and correspondingly, the prepared state is approximately the ground state for the final Hamiltonian.
We start by importing required libraries
using EaRyd
using CairoMakie
using EaRydPlots
using KrylovKit
using SparseArraysGround state properties
To get a sense of the quantum phase transition, let us first calculate the ground states' properties for Hamiltonians with different parameters. We will focus on ground state of 1D chain for simplicity. We consider a chain with 9 atoms, where nearby atoms are seperated by a distance of 5.72 $\mu m$. Such a system is generated by the following codes
nsites = 9
atoms = generate_sites(ChainLattice(), nsites, scale=5.72)
We fix the Rabi frequency to be $Ω = 2π * 4$, and study the ground state as a function of detuning $Δ$.
Ω = 2π * 4
Δ_step = 30
Δ = LinRange(-10*2π , 10*2π , Δ_step)30-element LinRange{Float64, Int64}:
-62.8319,-58.4986,-54.1654,-49.8322,…,49.8322,54.1654,58.4986,62.8319We compute the Rydberg density profile for each parameter of $\Delta$.
density_g = zeros(Δ_step, nsites)
for ii=1: Δ_step
h_ii = rydberg_h(atoms; Δ=Δ[ii], Ω)
h_m = SparseMatrixCSC(h_ii)
vals, vecs, info = KrylovKit.eigsolve(h_m, 1, :SR)
g_state = ArrayReg(vecs[1])
for jj in 1:nsites
density_g[ii, jj] = real(expect(put(nsites, jj=>Op.n), g_state))
end
endTo compare, we first plot the density profile when $\Delta= -10*2π$,
lines(1:nsites, density_g[1, :])
We can see that the Rydberg densities in this case is close to 0 for all sites. In contrast, for $\Delta= -10*2π$, the density shows clear $Z_2$ ordered profile
lines(1:nsites, density_g[20, :])
More generally, we can plot an order parameter as a function of $\Delta$ to clearly see the onset of phase transition. Such a parameter is defined by the difference of Rydberg densities in even and odd sites.
order_para = map(1: Δ_step) do ii
sum(density_g[ii, 1:2:8])- sum(density_g[ii, 2:2:8])
end
lines(collect(Δ)/2π, order_para)
From the density profile of ground states and the change of order parameter, we are convinced that there is indeed a phase transition taking place during the change of $\Delta$. Below we show that by slowly changing the parameters in the Hamiltonian, we can follow the trajectory of the ground states and adibatically evolve from all atoms being in state 0 to the $Z_2$ ordered state.
Ordered state prepration in 1D
We first specify the adiabatic pulse sequence for Rabi frequency by using the built-in waveform function piecewise_linear
total_time = 3.0
Ω_max = 2π * 4
Ω = piecewise_linear(clocks=[0.0, 0.1, 2.1, 2.2, total_time], values=[0.0, Ω_max, Ω_max, 0, 0])
draw(Ω)
#Δ1 = piecewise_linear(clocks=[0.0, 0.6, 2.1, 3], values=[-10*2π, -10*2π, 10*2π, 10* 2π])
The detuning sequence could also be created in a similar way
U1 = -2π * 10
U2 = 2π * 10
Δ = piecewise_linear(clocks=[0.0, 0.6, 2.1, total_time], values=[U1, U1, U2, U2])
draw(Δ)
We then generate the positions of a 1D atomic chain by using the function generate_sites
nsites = 9
atoms = generate_sites(ChainLattice(), nsites, scale=5.72)
Note that we have specify the nearest-neighbor atoms being seperated by 5.72 $\mu m$ in order to prepare a $Z_2$ ordered state. With the waveforms and atomic coordinate specified, the time-dependent Hamiltonian can be simply generated by the following
h = rydberg_h(atoms; Δ, Ω)Hamiltonian
Term 1
∑(n=1:9) Waveform(_, 3.0)(t)/2 σ^x
Term 2
-∑(n=1:9) Waveform(_, 3.0)(t) n
Term 3
∑(n=1:9) 5.42e6/|r_i - r_j|^6 n_i n_jWe then specify all atoms to be in ground state initially, and set up the emulation problem by choosing the ODE solver
reg = zero_state(9)
prob = SchrodingerProblem(reg, total_time, h; dt=1e-5, adaptive=false)
integrator = init(prob, Vern8())t: 0.0
u: 512-element Vector{ComplexF64}:
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0.0 + 0.0imWe measure the Rydberg density for each site and time step
densities = []
for _ in integrator
push!(densities, [expect(put(nsites, i=>Op.n), reg) for i in 1:nsites])
end
D = hcat(densities...)9×300000 Matrix{ComplexF64}:
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